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4 Sheaves of Modules, Vector Bundles, and (Quasi-)Coherent Sheaves

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4 Sheaves of Modules, Vector Bundles, and (Quasi-)Coherent Sheaves 4 Sheaves of modules, vector bundles, and (quasi-)coherent sheaves “If you believe a ring can be understood geometrically as functions its spec- trum, then modules help you by providing more functions with which to measure and characterize its spectrum.” – Andrew Critch, from MathOver- flow.net So far we discussed general properties of sheaves, in particular, of rings. Similar as in the module theory in abstract algebra, the notion of sheaves of modules allows us to increase our understanding of a given ringed space (or a scheme), and to provide further techniques to play with functions, or function-like objects. There are particularly important notions, namely, quasi-coherent and coherent sheaves. They are analogous notions of the usual modules (respectively, finitely generated modules) over a given ring. They also generalize the notion of vector bundles. Definition 38. Let (X, ) be a ringed space. A sheaf of -modules, or simply an OX OX -module, is a sheaf on X such that OX F (i) the group (U) is an (U)-module for each open set U X; F OX ✓ (ii) the restriction map (U) (V ) is compatible with the module structure via the F !F ring homomorphism (U) (V ). OX !OX A morphism of -modules is a morphism of sheaves such that the map (U) F!G OX F ! (U) is an (U)-module homomorphism for every open U X. G OX ✓ Example 39. Let (X, ) be a ringed space, , be -modules, and let ' : OX F G OX F!G be a morphism. Then ker ', im ', coker ' are again -modules. If is an - OX F 0 ✓F OX submodule, then the quotient sheaf / is an -module. Any direct sum, direct F F 0 OX product, direct limit, or inverse limit of -modules is an -module. OX OX Definition 40. Let , be two -modules. We denote the group of morphisms from F G OX to by Hom ( , ) (or HomX ( , ) or Hom( , ) if there is no confusing). F G OX F G F G F G If U X is an open subset, then is an -module. The presheaf ✓ F|U OX |U U Hom ( U , U ) 7! OX |U F| G| is indeed a sheaf, and we will call it the sheaf Hom. We denote it by om ( , ). H OX F G The sheaf associated to the presheaf U (U) (U) (U) is called the tensor product 7! F ⌦OX G . If there is no confusing, we write simply . F⌦OX G F⌦G Definition 41. An -module is free if it is isomorphic to a direct sum of copies OX F of .Itislocally free if X can be covered by open subsets U for which is a free OX F|U -module. In the case, the rank of is the number of free copies of the structure OX |U F sheaf needed. Note that the rank of a locally free sheaf is the same everywhere when X is connected. A locally free sheaf of rank 1 is called an invertible sheaf. A sheaf of ideals on X is a sheaf of modules I which is a subsheaf of , that is, I (U) OX is an ideal in (U) for each open set U X. OX ✓ 17 Topic 1 : Sheaf Example 42. Let A be a ring, and I A be an ideal. By the definition, the set V (I) ⇢ ✓ X =SpecA is a closed subset. We identify V (I) as an affine scheme Y =SpecA/I,so that we have a short exact sequence of sheaves of -modules OX 0 I 0 ! !OX !OY ! which is the natural analogue of 0 I A A/I 0. ! ! f ! ! Indeed, if there is a closed subscheme Y , X, then the map X f Y induces a ! O ! ⇤O surjection on its global sections A B. In other words, B A/I for some ideal I in A. ! ' It is straightforward that the quotient A A/I is compatible with every localization, ! hence, we may cover Y by D(f ) Y where D(f ) is a collection of distinguished open i \ { i } subsets of X =SpecA. In particular, Y has to coincide with the affine scheme Spec A/I. The ideal sheaf I is defined from local data on distinguished open sets. Namely, if U = D(s)wheres A, then we assign I (U):=I A . 2 s ✓ s Definition 43. Let f :(X, ) (Y, ) be a morphism of ringed spaces, let be OX ! OY F an X -module, and let be an Y -module. O G O # Note that the direct image f is an f X -module. Together with the morphism f : ⇤F ⇤O Y f X of sheaves of rings on Y , we have a natural Y -module structure on f . O ! ⇤O O ⇤F We call it the direct image of , or the pushforward of by f. F F Also note that the inverse image sheaf f 1 is an f 1 -module. Thanks to the adjoint − G − OY property of f 1, there is a unique morphism f 1 which corresponds f # : − − OY !OX Y f X . To provide an X -module structure, we take the tensor product O ! ⇤O O 1 f ⇤ := f − f 1 X G G⌦ − OY O and call it the inverse image of , or the pullback of by f. G G Remark 44. f and f ⇤ are adjoint functors between the category of X -modules and ⇤ O the category of -modules. OY If we restrict our focus to the case of schemes, we already have enough general notions and give the definition of quasi-coherent and coherent sheaves on schemes. For instance, on an affine scheme Spec A, a quasi-coherent (resp., coherent) sheaf is completely determined by an A-module (resp., a finitely generated A-module) M. If there is an A-module M, then it is compatible with every localization of A. In particular, for each s A,the 2 localized module Ms is an As-module and the module structure is compatible with the localization. This enables us to build a sheaf of -module. However, before we OSpec A proceed forward, let us recall the notion of vector bundles and see why it is natural to consider quasi-coherent and coherent sheaves instead of vector bundles. Let X be a real topological manifold. A vector bundle of rank r on X is defined as a E map ⇡ : X such that for every x X, there is an open neighborhood U of x and E! 2 r 1 a homeomorphism (also called a local trivialization) φU : U R ⇡− (U) satisfying r ⇥ ! 1 r (⇡ φU )(x, v)=x for any x U and v R . In particular, the fiber ⇡− (x) R has ◦ 2 2 ' the structure of a vector space. A rank 1 vector bundle is called a line bundle. 18 When we have two trivializations over U1 and U2, we have an isomorphism on the 1 r r intersection φ− φ :(U U ) (U U ) which sends (x, v)to(x, T (x)v) U2 U1 1 2 R 1 2 R 12 ◦ \ ⇥ ! \ ⇥ r for some GL(r)-valued function T12 : U1 U2 GL(r) = Aut(R ). When U3 is another \ ! open subset of X with a local trivialization of , then they satisfy the cocycle condition E T23 U1 U2 U3 T12 U1 U2 U3 = T13 U1 U2 U3 . | \ \ ◦ | \ \ | \ \ In fact, an open cover U of X with local trivializations φ for each i and transition { i} Ui functions Tij recover the vector bundle ⇡ : X by “gluing together” the various r { } E! Ui R along Ui Uj using Tij. (Compare with the gluing property of sheaves.) ⇥ \ Now we associate a sheaf to a vector bundle ⇡ : X, called the sheaf of sections. Note E! that a section of a vector bundle on an open subset U X is a function s : U E ✓ !E such that ⇡ s = id . Hence, if we take as the sheaf of continuous functions on X ◦ U OX and regard (X, ) as a ringed space, then the sheaf OX F U (U):= s : U sections of ⇡ over U 7! F { !E | } has an -module structure. In particular, when ⇡ admits a local trivialization over OX U, the sections over U naturally identify with the r-tuples of functions U R, and we r ! conclude that ⊕ . In other words, a vector bundle of rank r gives a locally F|U 'OX |U free sheaf of rank r on X. Let us write down a little more detail inside the above construction. Let U X be an ⇢ open set, and let Ui is an open covering of U by open subsets Ui equipped with trivi- { } 1 alizations. Let s : U be a section over U. The restriction s Ui : Ui ⇡− (Ui) onto !E r | ! each subset Ui induces a vector-valued function s ~ i : Ui R , via the local trivialization 1 1 r ! φ− : ⇡− (Ui) U R together with the second projection. Then we have Tijs ~ i = s ~ j: Ui ! ⇥ chase the following diagram 1 s U U φU− | i\ j 1 i r pr2 r Ui Uj / ⇡− (Ui Uj) / (Ui Uj) R / R \ \ \ ⇥ idU U Tij Tij i\ j ⇥ φ 1 s U U U− ✏ | i\ j 1 j r pr2 ✏ r Ui Uj / ⇡− (Ui Uj) / (Ui Uj) R / R \ \ \ ⇥ where the composition of the horizontal maps in the top row describes s ~ i, and the one for the bottom row describes s ~ . Hence, the locally free sheaf of sections of ⇡ : X j F E! provides the same transition functions T GL(r, (U U )), and of course, they ij 2 O i \ j satisfy the cocycle condition.
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